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Statistics

3.1 Terminology

Statistics3.1 Terminology
  1. Preface
  2. 1 Sampling and Data
    1. Introduction
    2. 1.1 Definitions of Statistics, Probability, and Key Terms
    3. 1.2 Data, Sampling, and Variation in Data and Sampling
    4. 1.3 Frequency, Frequency Tables, and Levels of Measurement
    5. 1.4 Experimental Design and Ethics
    6. 1.5 Data Collection Experiment
    7. 1.6 Sampling Experiment
    8. Key Terms
    9. Chapter Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  3. 2 Descriptive Statistics
    1. Introduction
    2. 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
    3. 2.2 Histograms, Frequency Polygons, and Time Series Graphs
    4. 2.3 Measures of the Location of the Data
    5. 2.4 Box Plots
    6. 2.5 Measures of the Center of the Data
    7. 2.6 Skewness and the Mean, Median, and Mode
    8. 2.7 Measures of the Spread of the Data
    9. 2.8 Descriptive Statistics
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  4. 3 Probability Topics
    1. Introduction
    2. 3.1 Terminology
    3. 3.2 Independent and Mutually Exclusive Events
    4. 3.3 Two Basic Rules of Probability
    5. 3.4 Contingency Tables
    6. 3.5 Tree and Venn Diagrams
    7. 3.6 Probability Topics
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Bringing It Together: Practice
    13. Homework
    14. Bringing It Together: Homework
    15. References
    16. Solutions
  5. 4 Discrete Random Variables
    1. Introduction
    2. 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
    3. 4.2 Mean or Expected Value and Standard Deviation
    4. 4.3 Binomial Distribution (Optional)
    5. 4.4 Geometric Distribution (Optional)
    6. 4.5 Hypergeometric Distribution (Optional)
    7. 4.6 Poisson Distribution (Optional)
    8. 4.7 Discrete Distribution (Playing Card Experiment)
    9. 4.8 Discrete Distribution (Lucky Dice Experiment)
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. References
    16. Solutions
  6. 5 Continuous Random Variables
    1. Introduction
    2. 5.1 Continuous Probability Functions
    3. 5.2 The Uniform Distribution
    4. 5.3 The Exponential Distribution (Optional)
    5. 5.4 Continuous Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  7. 6 The Normal Distribution
    1. Introduction
    2. 6.1 The Standard Normal Distribution
    3. 6.2 Using the Normal Distribution
    4. 6.3 Normal Distribution—Lap Times
    5. 6.4 Normal Distribution—Pinkie Length
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  8. 7 The Central Limit Theorem
    1. Introduction
    2. 7.1 The Central Limit Theorem for Sample Means (Averages)
    3. 7.2 The Central Limit Theorem for Sums (Optional)
    4. 7.3 Using the Central Limit Theorem
    5. 7.4 Central Limit Theorem (Pocket Change)
    6. 7.5 Central Limit Theorem (Cookie Recipes)
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  9. 8 Confidence Intervals
    1. Introduction
    2. 8.1 A Single Population Mean Using the Normal Distribution
    3. 8.2 A Single Population Mean Using the Student's t-Distribution
    4. 8.3 A Population Proportion
    5. 8.4 Confidence Interval (Home Costs)
    6. 8.5 Confidence Interval (Place of Birth)
    7. 8.6 Confidence Interval (Women's Heights)
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. Solutions
  10. 9 Hypothesis Testing with One Sample
    1. Introduction
    2. 9.1 Null and Alternative Hypotheses
    3. 9.2 Outcomes and the Type I and Type II Errors
    4. 9.3 Distribution Needed for Hypothesis Testing
    5. 9.4 Rare Events, the Sample, and the Decision and Conclusion
    6. 9.5 Additional Information and Full Hypothesis Test Examples
    7. 9.6 Hypothesis Testing of a Single Mean and Single Proportion
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. Solutions
  11. 10 Hypothesis Testing with Two Samples
    1. Introduction
    2. 10.1 Two Population Means with Unknown Standard Deviations
    3. 10.2 Two Population Means with Known Standard Deviations
    4. 10.3 Comparing Two Independent Population Proportions
    5. 10.4 Matched or Paired Samples (Optional)
    6. 10.5 Hypothesis Testing for Two Means and Two Proportions
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  12. 11 The Chi-Square Distribution
    1. Introduction
    2. 11.1 Facts About the Chi-Square Distribution
    3. 11.2 Goodness-of-Fit Test
    4. 11.3 Test of Independence
    5. 11.4 Test for Homogeneity
    6. 11.5 Comparison of the Chi-Square Tests
    7. 11.6 Test of a Single Variance
    8. 11.7 Lab 1: Chi-Square Goodness-of-Fit
    9. 11.8 Lab 2: Chi-Square Test of Independence
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  13. 12 Linear Regression and Correlation
    1. Introduction
    2. 12.1 Linear Equations
    3. 12.2 The Regression Equation
    4. 12.3 Testing the Significance of the Correlation Coefficient (Optional)
    5. 12.4 Prediction (Optional)
    6. 12.5 Outliers
    7. 12.6 Regression (Distance from School) (Optional)
    8. 12.7 Regression (Textbook Cost) (Optional)
    9. 12.8 Regression (Fuel Efficiency) (Optional)
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  14. 13 F Distribution and One-way Anova
    1. Introduction
    2. 13.1 One-Way ANOVA
    3. 13.2 The F Distribution and the F Ratio
    4. 13.3 Facts About the F Distribution
    5. 13.4 Test of Two Variances
    6. 13.5 Lab: One-Way ANOVA
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  15. A | Appendix A Review Exercises (Ch 3–13)
  16. B | Appendix B Practice Tests (1–4) and Final Exams
  17. C | Data Sets
  18. D | Group and Partner Projects
  19. E | Solution Sheets
  20. F | Mathematical Phrases, Symbols, and Formulas
  21. G | Notes for the TI-83, 83+, 84, 84+ Calculators
  22. H | Tables
  23. Index

Probability is a measure that is associated with how certain we are of results, or outcomes, of a particular activity. When the activity is a planned operation carried out under controlled conditions, it is called an experiment. If the result is not predetermined, then the experiment is said to be a chance experiment. Each time the experiment is attempted is called a trial.

Examples of chance experiments include the following:

  • flipping a fair coin,
  • spinning a spinner,
  • drawing a marble at random from a bag, and
  • rolling a pair of dice.

A result of an experiment is called an outcome. The sample space of an experiment is the set, or collection, of all possible outcomes.

There are four main ways to represent a sample space:

Flipping a Fair Coin Flipping Two Fair Coins
Systematic List of Outcomes heads (H)
tails (T)
HH
HT
TH
TT
Tree Diagram*
Table 3.1 Figure 3.2
Figure 3.3
Venn Diagram*
Figure 3.4
Figure 3.5
Set Notation S= { H, T } S= { H, T } S= { HH, HT, TH, TT } S= { HH, HT, TH, TT }

*We will investigate tree diagrams and Venn diagrams in Section 3.5.

Note—when represented as a set, the sample space is denoted with an uppercase S.

An event is any combination of outcomes. It is a subset of the sample space, so uppercase letters like A and B are commonly used to represent events. For example, if the experiment is to flip three fair coins, event A might be getting at most one head.

The probability of an event A is written P(A), and  0  P( A )   1.P(A) = 0  0  P( A )   1.P(A) = 0 means the event A can never happen. P(A) = 1 means the event A always happens. P(A) = 0.5 P(A) = 0.5 means the event A is equally likely to occur or not to occur.

Image shows a number line from zero to one with a tick and label at one half. Tick zero represents the probability of an impossible event. Tick one represents the probability of a certain event. Tick one half represents the probability of an event that is equally likely to happen or not. Above the number line, an arrow points from one half toward zero showing that as probability moves closer to zero events are less likely. An arrow points from one half toward one showing that as probability moves closer to one events are more likely.
Figure 3.6

If two outcomes or events are equally likely, then they have equal probability. For example, if you toss a fair, six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. If you toss a fair coin, a Head (H) and a Tail (T) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.

To calculate the probability of an event A when all outcomes in the sample space are equally likely, count the number of outcomes for event A and divide by the total number of outcomes in the sample space. This is known as the theoretical probability of A.

Theoretical Probability of Event A

P(A)= Number of outcomes in event A Total number of possible outcomes. P(A)= Number of outcomes in event A Total number of possible outcomes.

For example, if you toss a fair dime and a fair nickel, the sample space is {HH, TH, HT, TT} where T = tails and H = heads. The sample space has four outcomes. Let A represent the outcome getting one head. There are two outcomes that meet this condition {HT, TH}, so P(A)= 2 4 = 1 2 =.5. P(A)= 2 4 = 1 2 =.5.

Theoretical probability is not sufficient in all situations, however. Suppose we want to calculate the probability that a randomly selected car will run a red light at a given intersection. In this case, we need to look at events that have occurred, not theoretical possibilities. We could install a traffic camera and count the number of times that cars failed to stop when the light was red and the total number of cars that passed through the intersection for a period of time. These data will allow us to calculate the experimental, or empirical, probability that a car runs the red light.

Experimental Probability of Event A

P(A)= Number of times event A occurs. Total number of trials P(A)= Number of times event A occurs. Total number of trials

While theoretical and experimental methods provide two different ways to calculate probability, these methods are closely related. If you flip one fair coin, there is one way to obtain heads and two possible outcomes. So, the theoretical probability of heads is 1 2 1 2 . Probability does not predict short-term results, however. If an experiment involves flipping a coin 10 times, you should not expect exactly five heads and five tails. The probability of any outcome measures the long-term relative frequency of that outcome. If you continue to flip the coin (from 20 to 2,000 to 20,000 times) the relative frequency of heads approaches .5 (the probability of heads).This important characteristic of probability experiments is known as the law of large numbers, which states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed, or empirical, relative frequency will approach the theoretical probability.

Suppose you roll one fair, six-sided die with the numbers {1, 2, 3, 4, 5, 6} on its faces. Let event E = rolling a number that is at least five. There are two outcomes {5, 6}. P(E)= 2 6 . P(E)= 2 6 . If you were to roll the die only a few times, you would not be surprised if your observed results did not match the probability. If you were to roll the die a very large number of times, you would expect that, overall, 2 6 2 6 of the rolls would result in an outcome of at least five. You would not expect exactly 2 6 2 6 , but the long-term relative frequency of obtaining this result would approach the theoretical probability of 2 6 2 6 as the number of repetitions grows larger and larger.

It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair, or biased. Two math professors in Europe had their statistics students test the Belgian one-euro coin and discovered that in 250 trials, a head was obtained 56 percent of the time and a tail was obtained 44 percent of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later we will learn techniques to use to work with probabilities for events that are not equally likely.

OR EventAn outcome is in the event A OR B if the outcome is in A or is in B or is in both A and B. For example, let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. A OR B = {1, 2, 3, 4, 5, 6, 7, 8}. Notice that 4 and 5 are not listed twice.

AND EventAn outcome is in the event A AND B if the outcome is in both A and B at the same time. For example, let A and B be
{1, 2, 3, 4, 5} and {4, 5, 6, 7, 8}, respectively. Then A AND B = {4, 5}.

The complement of event A is denoted A′ (read "A prime"). A′ consists of all outcomes that are not in A. Notice that
P(A) + P(A′) = 1. For example, let S = {1, 2, 3, 4, 5, 6} and let A = {1, 2, 3, 4}. Then, A′ = {5, 6}. P(A) = 4646, P(A′) = 2626, and P(A) + P(A′) = 4 6 + 2 6 4 6 + 2 6 = 1.

The conditional probability of A given B is written P(A|B), read "the probability of A, given B." P(A|B) is the probability that event A will occur given that the event B has already occurred. A conditional probability reduces the sample space. We calculate the probability of A from the reduced sample space B. The formula to calculate P(A|B) is P(A|B) = P(A AND B) P(B) P(A AND B) P(B) where P(B) is greater than zero.

For example, suppose we toss one fair, six-sided die. The sample space S = {1, 2, 3, 4, 5, 6}. Let A = {2, 3} and B = {2, 4, 6}. P(A|B) represents the probability that a randomly selected outcome is in A given that it is in B. We know that the outcome must lie in B, so there are three possible outcomes. There is only one outcome in B that also lies in A, so P(A|B) = 1 3 1 3 .

We get the same result by using the formula. Remember that S has six outcomes.

P(A|B) = P(A AND B) P(B) = (the number of outcomes that are 2 or 3 and even in S) 6 (the number of outcomes that are even in S) 6 = 1 6 3 6 = 1 3 P(A AND B) P(B) = (the number of outcomes that are 2 or 3 and even in S) 6 (the number of outcomes that are even in S) 6 = 1 6 3 6 = 1 3

Understanding Terminology and SymbolsIt is important to read each problem carefully to think about and understand what the events are. Understanding the wording is the first very important step in solving probability problems. Reread the problem several times if necessary. Clearly identify the event of interest. Determine whether there is a condition stated in the wording that would indicate that the probability is conditional; carefully identify the condition, if any.

Example 3.1

The sample space S is the whole numbers starting at one and less than 20.

  1. S = ________

    Let event A = the even numbers and event B = numbers greater than 13.

  2. A = ________, B = ________
  3. P(A) = ________, P(B) = ________
  4. A AND B = ________, A OR B = ________
  5. P(A AND B) = ________, P(A OR B) = ________
  6. A′ = ________, P(A′) = ________
  7. P(A) + P(A′) = ________
  8. P(A|B) = ________, P(B|A) = ________; are the probabilities equal?
Try It 3.1

The sample space S is all the ordered pairs of two whole numbers, the first from one to three and the second from one to four (Example: (1, 4)).

  1. S = ________

    Let event A = the sum is even and event B = the first number is prime.
  2. A = ________, B = ________
  3. P(A) = ________, P(B) = ________
  4. A AND B = ________, A OR B = ________
  5. P(A AND B) = ________, P(A OR B) = ________
  6. B′ = ________, P(B′) = ________
  7. P(A) + P(A′) = ________
  8. P(A|B) = ________, P(B|A) = ________; are the probabilities equal?

Example 3.2

A fair, six-sided die is rolled. The sample space, S, is {1, 2, 3, 4, 5, 6}. Describe each event and calculate its probability.

  1. Event T = the outcome is two.
  2. Event A = the outcome is an even number.
  3. Event B = the outcome is less than four.
  4. The complement of A
  5. A GIVEN B
  6. B GIVEN A
  7. A AND B
  8. A OR B
  9. A OR B′
  10. Event N = the outcome is a prime number.
  11. Event I = the outcome is seven.

Example 3.3

Table 3.2 describes the distribution of a random sample S of 100 individuals, organized by gender and whether they are right or left-handed.

Right-Handed Left-Handed
Males 43 9
Females 44 4
Table 3.2

Let’s denote the events M = the subject is male, F = the subject is female, R = the subject is right-handed, L = the subject is left-handed. Compute the following probabilities:

  1. P(M)
  2. P(F)
  3. P(R)
  4. P(L)
  5. P(M AND R)
  6. P(F AND L)
  7. P(M OR F)
  8. P(M OR R)
  9. P(F OR L)
  10. P(M')
  11. P(R|M)
  12. P(F|L)
  13. P(L|F)
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